Self-similar Blow-Up Profiles for Slightly Supercritical Nonlinear Schrödinger Equations

Bahri, Yakine; Martel, Yvan; Raphaël, Pierre

doi:10.1007/s00023-020-01006-z

Cited by 2 publications

(1 citation statement)

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“…The strategy relies on the matching asymptotic argument. Following the idea in [2], we aim at finding directly the relation (1.8) such that we can glue the outgoing solution at r → +∞ which is a small deformation of the singular solution, with the rescaled smooth solution which bifurcates from of the Emden-Fowler solution. The matching asymptotic strategy was implemented to construct special solutions like ground states (see [5,7]) or self-similar profiles for the energy supercritical nonlinear heat equation(see [6]).…”

Section: Introductionmentioning

confidence: 99%

Infinitely many positive solutions of a Gross-Pitaevskii equation in the presence of a harmonic potential and combined nonlinearities

Bahri¹,

Hajaiej²

2022

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The main goal of this paper is to address an important conjecture in the field of differential equations in the presence of a harmonic potential. While in the subcritical case, the uniqueness of positive solution has been addressed by Hirose and Ohta in [12] in 2007, the problem has remained open for years in the supercritical case. In [9], the author obtained interesting numerical computations suggesting that for some bifurcating parameter λ, the equation has many positive solutions that vanish at infinity. In this paper, we provide a proof to this claim by constructing an accountable number of solutions that bifurcate from the unique singular solutions with λ close to the first eigenvalue λ 1 of the harmonic operator −∆ + |x| 2 . Our method hinges on a matching argument, and applies to the supercritical case, and to the supercritical case in the presence of a subcritical, critical or supercritical perturbation.

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